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Keywords:
distance spectrum; distance characteristic polynomial; $D$-spectrum determined by its $D$-spectrum
distance spectrum; distance characteristic polynomial; $D$-spectrum determined by its $D$-spectrum
Summary:
Let $G$ be a connected graph with vertex set $V(G)=\{v_{1},v_{2},\ldots ,v_{n}\}$. The distance matrix $D(G)=(d_{ij})_{n\times n}$ is the matrix indexed by the vertices of\/ $G$, where $d_{ij}$ denotes the distance between the vertices $v_{i}$ and $v_{j}$. Suppose that $\lambda _{1}(D)\geq \lambda _{2}(D)\geq \nobreak \cdots \geq \lambda _{n}(D)$ are the distance spectrum of $G$. The graph $G$ is said to be determined by its \hbox {$D$-spectrum} if with respect to the distance matrix $D(G)$, any graph having the same spectrum as $G$ is isomorphic to $G$. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their $D$-spectra.
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